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Preface xi To the Student xxi 0 PARAMETRIC EQUATIONS AND POAR COORD ATES 620 0.1 Curves Defined by Parametric Equations 621 Laboratory Project Running Circles around Circles 629 10.2 Calculus with Parametric Curves 630 Laboratory Project - Bizier Curves 639 10.3 Polar Coordinates 639 10.4 Areas and Lengths in Polar Coordinates 650 10.5 Conic Sections 654 10.6 Conic Sections in Polar Coordinates 662 Review 669 Problems Plus 672 I INFINITE SEQUENCES AND SERIES 674 11.1 Sequences 675 Laboratory Project - Logistic Sequences 687 11.2 Series 687 11.3 The Integral Test and Estimates of Sums 697 1.4 The Comparison Tests 705 11.5 Alternating Series 710 11.6 Absolute Convergence and the Ratio and Root Tests 714 11.7 Strategy for Testing Series 72 11.8 Power Series 723 S1.9 Representations of Functions as Power Series 728 S110 Taylor and Maclaurin Series 734 Laboratory Project An Elusive Limit 748 Writing Project- How Newton Discovered the Binomial Series 748 11. 11 Applications of Taylor Polynomials 749 Applied Project - Radiation from the Stars 757 Review 758 Problems Plus 761 12 VECTORS AND THE GEOMETRY OF SPACE 764 12.1 Three-Dimensional Coordinate Systems 765 12.2 Vectors 770 "12.3 The Dot Product 779 12.4 The Cross Product 786 Discovery Project - The Geometry of a Tetrahedron 794 12.5 Equations of Lines and Planes 794 Laboratory Project - Putting 3D in Perspective 804 12.6 Cylinders and Quadric Surfaces 804 Review 812 Problems Plus 815 --33, VECTOR FUNCTONS 816 13.1 Vector Functions and Space Curves 817 13.2 Derivatives and Integrals of Vector Functions 824 13.3 Arc Length and Curvature 830 13.4 Motion in Space: Velocity and Acceleration 838 Applied Project Kepler's Laws 848 Review 849 Problems Plus 852 114 I PARTIAL DERIVATIVES 854 14.1 Functions of Several Variables 855 14.2 Limits and Continuity 870 14.3 Partial Derivatives 878 14.4 Tangent Planes and Linear Approximations 892 14.5 The Chain Rule 901 14.6 Directional Derivatives and the Gradient Vector 910 14.7 Maximum and Minimum Values 922 Applied Project - Designing a Dumpster 933 Discovery Project - Quadratic Approximations and Critical Points 933 14.8 Lagrange Multipliers 934 Applied Project - Rocket Science 941 Applied Project - Hydro-Turbine Optimization 943 Review 944 Problems Plus 948 I5 MULTIPLE INTEGRALS 950 15.1 Double Integrals over Rectangles 951 15.2 Iterated Integrals 959 15.3 Double Integrals over General Regions 965 15.4 Double Integrals in Polar Coordinates 974 15.5 Applications of Double Integrals 980 15.6 Triple Integrals 990 Discovery Project. Volumes of Hyperspheres 1000 15.7 Triple Integrals in Cylindrical Coordinates 1000 Discovery Project * The Intersection of Three Cylinders 1005 15.8 Triple Integrals in Spherical Coordinates 1005 Applied Project Roller Derby 1012 15.9 Change of Variables in Multiple Integrals 1012 Review 1021 Problems Plus 024 J 16 VECTOR CALCULUS 1026 16.1 Vector Fields 1027 16.2 Line Integrals 1034 16.3 The Fundamental Theorem for Line Integrals 1046 16.4 Green's Theorem 1055 16.5 Curl and Divergence 1061 16.6 Parametric Surfaces and Their Areas 1070 16.7 Surface Integrals 1081 16.8 Stokes' Theorem 1092 Writing Project Three Men and Two Theorems 1098 16.9 The Divergence Theorem 1099 16.10 Summary 1105 Review 1106 Probiems Plus 11 09 - 7 SECOND-ORDER DIFFERENTIAL EQUATIONS 1 10 17.1 Second-Order Linear Equations 1111 17.2 Nonhomogeneous Linear Equations 1117 17.3 Applications of Second-Order Differential Equations 1125 17.4 Series Solutions 1133 Review 1137 "- APPENDIXES Al F Proofs of Theorems A2 H Complex Numbers A5 I Answers to Odd-Numbered Exercises A13